~~NOTOC~~
====== Angular Dependent Potentials ======
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The angular dependent potentials were developed for the Fe-Ni system. It is a generalization of the EAM model for the simulation of the covalent component of bonding.

The angular dependent potential (ADP) for [[http://imd.itap.physik.uni-stuttgart.de/|IMD]] was implemented by Franz Gähler. It was adapted for use in //potfit// by Daniel Schopf.

=====  Basic Theory  =====

The angular dependent potenial model was suggested by Mishin et al.((Mishin, Y., Mehl, M. J., and Papaconstantopoulos, //D. A. Acta Mater.//,** 53** (15), 4029-4041, (2005) )).
Since it is based on the EAM potential model, the first two terms in the expression for the energy are exactly the same as for EAM potentials:

$$E_{\text{total}}=\frac{1}{2}\sum_{i,j(j\neq i)}^N\Phi_{ij}(r_{ij})+\sum_iF_i(n_i)+\frac{1}{2}\sum_{i,\alpha}(\mu_i^\alpha)^2+\frac{1}{2}\sum_{i,\alpha,\beta}(\lambda_i^{\alpha\beta})^2-\frac{1}{6}\sum_i\nu_i^2$$

Here the indices $i$ and $j$ enumerate atoms and the superscripts $\alpha,\beta=1,2,3$ refer to the Cartesion directions. The first two terms are explained in detail on the [[EAM|EAM]] page. 

The additional three terms introduce non-central components of bonding through the vectors

$$\mu_i^\alpha = \sum_{j\neq i} u_{ij}(r_{ij})r_{ij}^\alpha$$

and tensors

$$\lambda_i^{\alpha\beta} = \sum_{j\neq i}w_{ij}(r_{ij})r_{ij}^\alpha r_{ij}^\beta$$

The quantities $\nu_i$ are traces of the $\lambda$-tensor:

$$\nu_i = \sum_\alpha\lambda_i^{\alpha\alpha}$$

These additional terms can be thought of as measures of the dipole ($\mu$) and quadrupole ($\lambda$) distortions of the local environment of an atom.

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

To describe a system with //N// atom types you need //N//(3//N//+7)/2 potentials.

^ # atom types ^ $\Phi_{ij}$ ^ $\rho_j$ ^ $F_i$ ^ $u_{ij}$ ^ $w_{ij}$ ^ Total # potentials ^
^ $N$ ^ $N(N+1)/2$ ^ $N$ ^ $N$ ^ $N(N+1)/2$ ^ $N(N+1)/2$ ^ $N(3N+7)/2$ ^
| 1 | 1 | 1 | 1 | 1 | 1 | 5 |
| 2 | 3 | 2 | 2 | 3 | 3 | 11 |
| 3 | 6 | 3 | 3 | 6 | 6 | 24 |
| 4 | 10 | 4 | 4 | 10 | 10 | 38 |

=====  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:

$\Phi_{00}, \ldots, \Phi_{0N}, \Phi_{11}, \ldots, \Phi_{1N}, \ldots, \Phi_{NN}$\\
$\rho_0, \ldots, \rho_N$\\
$F_0, \ldots, F_N$\\
$u_{00}, \ldots, u_{0N}, u_{11}, \ldots, u_{1N}, \ldots, u_{NN}$\\
$w_{00}, \ldots, w_{0N}, w_{11}, \ldots, w_{1N}, \ldots, w_{NN}$\\

=====  Special remarks  =====

Tabulated ADP potentials require the embedding function $F_i$ to be defined at a density of $1.0$.
This is necessary to fix the gauge degrees of freedom.