Bond-Order Potentials

Bond-order potentials aim to capture the effect of the nearest-neighbor environment on the behavior of any bond. Such potentials began with the work of Abell [30]. Here I only very lightly gloss over this very deep field of research. In the bond-order formalism, the total potential energy due to covalent bonds is:

$$U = \sum_i{\sum_{j>i} {\phi_{ij}}},$$ (276)

where the bond energy $\phi_{ij}$ between atoms $i$ and $j$ has repulsive and attractive components:

$$\phi_{ij} = V_R(r_{ij}) - \overline{b}_{ij}V_A(r_{ij}) ,$$ (277)

where $V_R$ and $V_A$ are Morse-type pair potentials:

$$V_R(r_{ij}) = f_{ij}(r_{ij})A_{ij}\exp(-\lambda_{ij}r_{ij}) {\rm\hspace{5mm} and}$$ (278)

$$V_A(r_{ij}) = f_{ij}(r_{ij})B_{ij}\exp(-\mu_{ij}r_{ij})$$ (279)

Here, $r_{ij}$ is the scalar separation between atoms $i$ and $j$. $A_{ij}$, $B_{ij}$, $\lambda_{ij}$, and $\mu_{ij}$ are all parameters specific to the two elements participating in the bond. (I use the following $ij$-subscript convention: when $ij$ appears on a variable, $i$ and $j$ refer to the individual atom indices; when $ij$ appears on a parameter or function, $i$ and $j$ are specific only to the elements of atoms $i$ and $j$.) The cutoff function $f_{ij}$ decays smoothly from 1 at some “inner” radius to 0 at some “outer” radius.

The bond order $\overline{b_{ij}}$ models all of the many-body chemistry:

$$\overline{b_{ij}} = {1\over2} \left[b_{ij}+b_{ji}+\mbox{corrections}\right],$$ (280)

where $b_{ij}$ is the contribution of atoms that neighbor atom $i$ to the bond order of the $ij$ bond. These involve complicated 3- and 4-body interactions. The corrections arise from the need to expand set of thermochemical data which the potentials are fit against.

Bond-order potentials first applied to metals, but expanded into silicon and hydrocarbons with the work of and Tersoff [31]and Brenner [32], respectively, producing what is called the “reactive empirical bond order potential (REBO). The more recent adaptive intermolecular reactive empirical bond-order (AIREBO) potential combines REBO with Lennard-Jones interactions and specific torsional potentials for better modeling of hydrocarbon chains. [33]

Polarizable versions of reactive potentials have also been developed. The charge-optimized many-body (COMB) potential is an extension of REBO in which the charge on each atom is allowed to change according to energies dictated by input parameters such as atom electronegativity. [34] Adding oxygen to the AIREBO hydrocarbon potential also necessitated including polarizability, leading to the qAIREBO potential. [35]