Editing Molecular dynamics (section)

=== Pair potentials versus many-body potentials ===

The potential functions representing the non-bonded energy are formulated as a sum over interactions between the particles of the system. The simplest choice, employed in many popular [[force field (chemistry)|force fields]], is the "pair potential", in which the total potential energy can be calculated from the sum of energy contributions between pairs of atoms. Therefore, these force fields are also called "additive force fields". An example of such a pair potential is the non-bonded [[Lennard-Jones potential]] (also termed the 6–12 potential), used for calculating van der Waals forces.

:<math>
U(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]
</math>

Another example is the Born (ionic) model of the ionic lattice. The first term in the next equation is [[Coulomb's law]] for a pair of ions, the second term is the short-range repulsion explained by Pauli's exclusion principle and the final term is the dispersion interaction term. Usually, a simulation only includes the dipolar term, although sometimes the quadrupolar term is also included.<ref>{{cite journal | vauthors = Cruz FJ, Lopes JN, Calado JC, Minas da Piedade ME | title = A molecular dynamics study of the thermodynamic properties of calcium apatites. 1. Hexagonal phases | journal = The Journal of Physical Chemistry B | volume = 109 | issue = 51 | pages = 24473–24479 | date = December 2005 | pmid = 16375450 | doi = 10.1021/jp054304p }}</ref><ref>{{cite journal | vauthors = Cruz FJ, Lopes JN, Calado JC |title=Molecular dynamics simulations of molten calcium hydroxyapatite |journal=Fluid Phase Equilibria |date=March 2006 |volume=241 |issue=1–2 |pages=51–58 |doi=10.1016/j.fluid.2005.12.021 |bibcode=2006FlPEq.241...51C }}</ref> When ''n<sub>l</sub>'' = 6, this potential is also called the [[Buckingham potential|Coulomb–Buckingham potential]].

:<math>U_{ij}(r_{ij}) = \frac {z_i z_j}{4 \pi \epsilon_0} \frac {1}{r_{ij}} + A_l \exp \frac {-r_{ij}}{p_l} + C_l r_{ij}^{-n_l} + \cdots
</math>

In [[Many-body problem|many-body potentials]], the potential energy includes the effects of three or more particles interacting with each other.<ref name="ReferenceA">{{cite journal| vauthors = Justo JF, Bazant MZ, Kaxiras E, Bulatov VV, Yip S |title=Interatomic potential for silicon defects and disordered phases|journal=Phys. Rev. B|date=1998|volume=58|issue=5|pages=2539–2550|doi=10.1103/PhysRevB.58.2539|arxiv= cond-mat/9712058 |bibcode= 1998PhRvB..58.2539J|s2cid=14585375}}</ref> In simulations with pairwise potentials, global interactions in the system also exist, but they occur only through pairwise terms. In many-body potentials, the potential energy cannot be found by a sum over pairs of atoms, as these interactions are calculated explicitly as a combination of higher-order terms. In the statistical view, the dependency between the variables cannot in general be expressed using only pairwise products of the degrees of freedom. For example, the [[Tersoff potential]],<ref>{{cite journal | vauthors = Tersoff J | title = Modeling solid-state chemistry: Interatomic potentials for multicomponent systems | journal = Physical Review B| volume = 39 | issue = 8 | pages = 5566–5568 | date = March 1989 | pmid = 9948964 | doi = 10.1103/physrevb.39.5566 | bibcode = 1989PhRvB..39.5566T }}</ref> which was originally used to simulate [[carbon]], [[silicon]], and [[germanium]], and has since been used for a wide range of other materials, involves a sum over groups of three atoms, with the angles between the atoms being an important factor in the potential. Other examples are the [[embedded atom model|embedded-atom method]] (EAM),<ref>{{cite journal | vauthors = Daw MS, Foiles SM, Baskes MI |author1-link=Murray S. Daw|author2-link=Stephen M. Foiles|author3-link=Michael Baskes |title=The embedded-atom method: a review of theory and applications |journal=Materials Science Reports |date=March 1993 |volume=9 |issue=7–8 |pages=251–310 |doi=10.1016/0920-2307(93)90001-U |url=https://zenodo.org/record/1258631 |doi-access=free }}</ref> the EDIP,<ref name="ReferenceA"/> and the Tight-Binding Second Moment Approximation (TBSMA) potentials,<ref>{{cite journal | vauthors = Cleri F, Rosato V | title = Tight-binding potentials for transition metals and alloys | journal = Physical Review B | volume = 48 | issue = 1 | pages = 22–33 | date = July 1993 | pmid = 10006745 | doi = 10.1103/physrevb.48.22 | bibcode = 1993PhRvB..48...22C }}</ref> where the electron density of states in the region of an atom is calculated from a sum of contributions from surrounding atoms, and the potential energy contribution is then a function of this sum.