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# Angular Dependent Potentials

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 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.1). 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 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(3N+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.

1)
Mishin, Y., Mehl, M. J., and Papaconstantopoulos, D. A. Acta Mater., 53 (15), 4029-4041, (2005)