~~NOTOC~~
====== Angular Pair Potentials ======
----

The angular pair potentials are a combination of the regular pair potentials and a angular dependent term similar to MEAM:

$$
V=\frac{1}{2}\sum\phi_{ij}(r_{ij})+\frac{1}{2}\sum f_{ij}(r_{ij})f_{ik}(r_{ik})g_i(\cos(\theta_{ijk}))
$$

With this approach it is possible to fit potentials such as this one [[http://www.sciencedirect.com/science/article/pii/S0022309396006047|here]].

Or it can be used to consider non-bonded angular interactions as DL_POLY does for the three-body terms (see 'tbp' interaction, p. 174 in the [[http://goo.gl/3ns4i0|manual]]). This can be achieved defining the $f_{ij} = 1$ between the pairs that will form 'triple-bonds' and with their cutoff controlling the central radius to account for angular bonded neighbours. Then, just choosing the $g$ function as an harmonic one.

===== Number of potential functions =====

To describe a system of $N$ atom types you need $N(N+2)$ potentials.

^ # atom types ^ $\phi_{ij}$ ^ $f_{ij}$ ^ $g_i$ ^ Total # potentials ^
^ $N$ ^ $N(N+1)/2$ ^ $N(N+1)/2$ ^ $N$ ^ $N(N+2)$ ^
| 1 | 1 | 1 | 1 | 3 |
| 2 | 3 | 3 | 2 | 8 |
| 3 | 6 | 6 | 3 | 15 |
| 4 | 10 | 10 | 4 | 24 |

=====  Order of potential functions  =====

The potential table is assumed to be symmetric, i.e. the potential for the atom types 1-0 is the same as the potential 0-1.

The order of the potentials in the potential file for $N$ atom types is:

$\phi_{00}, \ldots, \phi_{0N}, \phi_{11}, \ldots, \phi_{1N}, \ldots, \phi_{NN}$\\
$f_{00}, \ldots, f_{0N}, f_{11}, \ldots, f_{1N}, \ldots, f_{NN}$\\
$g_0, \ldots, g_N$