~~NOTOC~~
====== Two band EAM Potentials ======
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=====  Basic Theory  =====

The two band EAM potentials are an extension to the regular EAM potentials. Instead
of single transfer and embedding functions, different functions to model two bands are used.
Usually they are reffered to as d- and s-band contributions.

$$E_\text{total}=\frac{1}{2}\sum_{i<j}^N\Phi_{ij}(r_{ij})+\sum_iF_i^d(n_i^d)+\sum_iF_i^s(n_i^s)$$
where
$$n^d_i=\sum_{j\neq i}\rho^d_j(r_{ij}) \qquad \text{and} \qquad n^s_i=\sum_{j\neq i}\rho^s_j(r_{ij})$$

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

To describe a system with $N$ atom types you need $N(N+9)/2$ potentials.

^ # atom types ^ $\Phi_{ij}$ ^ $\rho^d_j$ ^ $F^d_i$ ^ $\rho^s_j$ ^ $F^s_i$ ^Total # potentials ^
^ $N$ ^ $N(N+1)/2$ ^ $N$ ^ $N$ ^ $N$ ^ $N$ ^ $N(N+9)/2$ ^
| 1 | 1 | 1 | 1 | 1 | 1 | 5 |
| 2 | 3 | 2 | 2 | 2 | 2 | 11 |
| 3 | 6 | 3 | 3 | 3 | 3 | 18 |
| 4 | 10 | 4 | 4 | 4 | 4 | 26 |

=====  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 TBEAM potentials in the potential file for //N// atom types is:

$\Phi_{00}, \ldots, \Phi_{0N}, \Phi_{11}, \ldots, \Phi_{1N}, \ldots, \Phi_{NN},$\\
$\rho^d_0, \ldots, \rho^d_N,$\\
$F^d_0, \ldots, F^d_N,$\\
$\rho^s_0, \ldots, \rho^s_N,$\\
$F^s_0, \ldots, F^s_N,$

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

Tabulated two band EAM 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.