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====== modified Tersoff Potential ======
----

The modified Tersoff potential((T. Kumagai et al., //Comp. Mat. Sci.// **39**, 457 (2007) )) 
is available when //potfit// is compiled with the ''tersoffmod'' 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

$$b_{ij} = \left(1+\zeta_{ij}^{\eta}\right)^{-\delta},$$

$$\zeta_{ij} = \sum_{k\neq i,j}f_c(r_{ik})g(\theta_{ijk})\exp[\alpha(r_{ij}-r_{ik})^\beta]$$

and 

$$ g(\theta_{ijk}) = c_1 + g_o(\theta)g_a(\theta)$$

$$g_o(\theta) = \frac{c_2(h-\cos\theta)^2}{c_3+(h-\cos\theta)^2}$$

$$g_a(\theta) = 1 + c_4\exp[-c_5(h-\cos\theta)^2].$$

The cutoff function $f_c$ is given as

$$ f_c(r_{ij}) = \begin{cases}
1 & r_{ij} \le R_1 \\
\frac{1}{2} + \frac{9}{16}\cos\left(\frac{\pi(r_{ij}-R_1)}{R_2-R_1}\right)-\frac{1}{16}
\cos\left(\frac{3\pi(r_{ij}-R_1)}{R_2-R_1}\right) & R_1 < r_{ij} < R_2 \\
0 & R_2 \le r_{ij}
\end{cases}.$$

==== Analytic functions ====

To defined a modified Tersoff potential in //potfit//, there is the special ''tersoff_mod_pot'' function
which requires 16 parameters.

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

While basically systems with more than one atom type can be described using modified Tersoff potentials,
this is completely untested. A system containing $N$ atom types requires $N(N+1)/2$ potential functions.

^ # atom types ^ tersoff_mod_pot ^
^ $N$ ^ $N(N-1)/2$ ^
| 1 | 1 | 
| 2 | 3 | 
| 3 | 6 | 
| 4 | 10 |

=====  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:

$\Phi_{00}, \ldots, \Phi_{0N}, \Phi_{11}, \ldots, \Phi_{1N}, \ldots, \Phi_{NN}$

==== IMD output ====

An analytic IMD potential which can be used with the ''tersoffmod2'' option of IMD is written to ''*.imd.tersoffmod.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.tersoffmod''.