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 (number of bonds and possibly also angles and distances) of an atom. This concept is based on the Linus Pauling bond order concept , and can be written in the form
This means that the potential is written as a simple pair potential depending on the distance between two atoms , but the strength of this bond is modified by the environment of the atom i via the term. Alternatively, the energy can be written in the form
where is an 'electron density' at the location of atom i. These two forms for the energy can be shown to be equivalent.
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
The original bond order potential concept has been developed further to include distinct bond orders for sigma bonds and pi bonds in the so called BOP potentials. Also the ReaxFF potential can be considered a bond-order potential although the motivation of its bond order terms is different from that described here.