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~~NOTOC~~
====== Tersoff Potentials ======

----

The Tersoff potential((J. Tersoff, //Phys. Rev. B// **39**, 5566 (1989) )) 
is available when //potfit// is compiled with the ''tersoff'' flag, which also implies 
the ''apot'' flag for analytic potentials.

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

The total potential energy is defined as

$$E_{\text{total}}=
\frac{1}{2}\sum_{i,j}f_c(r_{ij})\left(A_{ij}e^{-\lambda_{ij}r_{ij}}+b_{ij}B_{ij}
e^{-\mu_{ij}r_{ij}}\right)$$

where

$$b_{ij} = \chi_{ij}\left(1+\gamma_{ij}^{n_i}\zeta_{ij}^{n_i}\right)^{-\frac{1}{2n_i}},$$

$$\zeta_{ij} = \sum_{k\neq i,j}f_c(r_{ik})\omega_{ik}g(\theta_{ijk})$$

and 

$$ g(\theta_{ijk}) = 1 + \frac{c_{ij}^2}{d_{ij}^2}-\frac{c_{ij}^2}{d_{ij}^2+(h_{ij}-\cos\theta_{ijk})^2}.$$

The cutoff function $f_c$ is given as

$$ f_c(r_{ij}) = \begin{cases}
1 & r_{ij} \le R_{ij} \\
\frac{1}{2} + \frac{1}{2}\cos\left(\frac{\pi(r_{ij}-R_{ij})}{S_{ij}-R_{ij}}\right) & R_{ij} < r_{ij} < S_{ij} \\
0 & S_{ij} \le r_{ij}
\end{cases}.$$

==== Analytic functions ====

To defined an analytic Tersoff potential in //potfit//, there are 2 functional forms required. They
are available as the ''tersoff_pot'' and ''tersoff_mix'' function types. Out of the 13 free parameters
for each interaction, the first eleven are specified in the ''tersoff_pot'' potential. The other two, which
are only relevant for mixing potentials, are defined in the ''tersoff_mix'' potential. 

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

To describe a system with $N$ atom types you need $N^2$ potentials. They are not directly related to
functions in the potential energy, they are divided into pure and mixing potentials. For more details 
take a look at the examples.

^ # atom types ^ tersoff_pot ^ tersoff_mix ^ Total # potentials ^
^ $N$ ^ $N(N+1)/2$ ^ $N(N-1)/2$ ^ $N^2$ ^
| 1 | 1 | 0 | 1 |
| 2 | 3 | 1 | 4 |
| 3 | 6 | 3 | 9 |
| 4 | 10 | 6 | 16 |

=====  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:

$P_{00}, \ldots, P_{0N}, P_{11}, \ldots, P_{1N}, \ldots, P_{NN}$\\
$M_{01}, \ldots, M_{0N}, M_{12}, \ldots, M_{1N}, \ldots, M_{N-1,N}$

where $P$ stands for ''tersoff_pot'' and $M$ for ''tersoff_mix'' potentials.

==== IMD output ====

An analytic IMD potential which can be used with the ''tersoff2'' option of IMD is written to ''*.imd.tersoff.pot''.
This file, however, may not be used as a potential file for IMD. Instead, its contents need to be copied
INTO the IMD parameter file.

==== LAMMPS output ====

The potential can also be written in LAMMPS format. The name of the output file is ''*.lammps.tersoff''.