Model driver for Tersoff-style potentials ported from LAMMPS v004

Tobias Brink; Aidan P. Thompson; David E. Farrell; Mingjian Wen; Jerry Tersoff; J. Nord; Karsten Albe; Paul Erhart; Kai Nordlund; James F. Ziegler; Jochen P. Biersack; U. Littmark

doi:10.25950/57484673

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Tersoff_LAMMPS__MD_077075034781_004

There is a newer version of this item. See Tersoff_LAMMPS__MD_077075034781_005

Title A single sentence description.	Model driver for Tersoff-style potentials ported from LAMMPS v004
Description	Model driver that supports parametrizations of the Tersoff-style three-body potential. It is ported from LAMMPS and supports the same format for parameter files. The commonly used variants by Tersoff and Nord/Albe/Erhart/Nordlund are supported, see source citations for details. Optionally, a Ziegler-Biersack-Littmark potential for short interatomic distances can be smoothly joined via a Fermi function. This functionality mirrors the tersoff/zbl variant in LAMMPS.
Disclaimer A statement of applicability provided by the contributor, informing users of the intended use of this KIM Item.	None
Content Origin	Ported from LAMMPS

Contributor	Tobias Brink
Maintainer	Tobias Brink
Implementer	Tobias Brink Aidan P. Thompson David E. Farrell Mingjian Wen
Developer	Jerry Tersoff J. Nord Karsten Albe Paul Erhart Kai Nordlund James F. Ziegler Jochen P. Biersack U. Littmark
Published on KIM	2021
How to Cite	Click here to download this citation in BibTeX format.
Funding	Not available

Short KIM ID The unique KIM identifier code.	MD_077075034781_004
Extended KIM ID The long form of the KIM ID including a human readable prefix (100 characters max), two underscores, and the Short KIM ID. Extended KIM IDs can only contain alpha-numeric characters (letters and digits) and underscores and must begin with a letter.	Tersoff_LAMMPS__MD_077075034781_004
DOI	10.25950/57484673 https://doi.org/10.25950/57484673 https://commons.datacite.org/doi.org/10.25950/57484673
KIM Item Type	Model Driver
KIM API Version	2.2
Programming Language(s) The programming languages used in the code and the percentage of the code written in each one.	86.44% C++ 13.56% Python

Previous Version	Tersoff_LAMMPS__MD_077075034781_003

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## Description
This **Model Driver** utilizes the Tersoff potential, originally used for Silicon in its seminal publication by Tersoff in 1988. It can be used for both the T2 and T3 bond order models, which are simply different parameter sets for the form given below.

## Parameters
Symbols used in the model (taking into account $$\alpha_{ij}$$ is set to zero for this implementation of the model, and the modifications given in the note at the bottom of this documentation):

$$ m, \gamma_{ijk}, \lambda_3, c, d, cos\theta_0, n, \beta, \lambda_2, B, R, D, \lambda_1, A $$

These correspond to the following variables in the code:

$$ \text{m, gamma, lambda3 (1/distance units), c, d, costheta0 (must be a valid value between -1 & 1), n,  beta, lambda2 (1/distance units), B (energy units), Rc (distance units), Dc (distance units), lambda1 (1/distance units), & A (energy units).}  $$

The parameters n, beta, lambda2, B, lambda1, & A are used for two-body interactions, while m, gamma, lambda3, c, d, costheta0 are only used for three-body interactions.

Additionally, the code requires three other variables to determine the elements being used:

$$ \text{Element 1}: \text{The center atom in a 3-body interaction} $$

$$ \text{Element 2}: \text{The atom bonded to the center atom} $$

$$ \text{Element 3}: \text{The atom influencing the 1-2 bond in a bond-order sense} $$

## Details
The interatomic potential is given as:

$$ E = \sum_i E_i =  {1 \over 2}{\sum_{i \neq j}V_{ij}}$$

Where $$E$$ is the total energy of the system, decomposed into site energy, $$E_i$$, and bond energy, $$V_{ij}$$. The indices $$i$$ and $$j$$ run through all atoms in the system, and $$r_{ij}$$ is the distance from atom $$i$$ to atom $$j$$.

$$V_{ij}$$ has the form:

$$ V_{ij} =  f_c(r_{ij})[a_{ij}f_R(r_{ij})+b_{ij}f_A(r_{ij})] $$

The functions $$f_A$$ and $$f_R$$ are the respective attractive and repulsive pair potentials:

$$ f_A = Ae^{-\lambda_1r} $$

$$ f_R = -Be^{-\lambda_2r} $$

Moreover, $$f_c$$ is the cutoff function with the form:

$$ f_c(r) = \begin{cases}
      1 &  \text{if  } r < R-D\\
      {1 \over 2}-{1 \over 2}sin[{\pi \over 2}{(r-R) \over D}]& \text{if  } R-D < r < R+D\\
      0 & \text{if  } r > R+D
    \end{cases} $$

The bond-order function, $$b_{ij}$$, is given as:

$$b_{ij} = (1+\beta^n\zeta^n_{ij})^{-1 / 2n} $$

Where $$\zeta_{ij}$$ is the coordination function, which gives the bond coordination of the environment:

$$ \zeta_{ij} = \sum_{k \neq i,j}f_c(r_{ik})g(\theta_{ijk})\exp[\lambda_3^3(r_{ij}-r_{ik})^3] $$

$$g(\theta_{ijk})$$ is as follows:

$$ g(\theta_{ijk}) =  1+ {c^2 \over d^2}- {c^2 \over [d^2+(h -cos\theta)^2]} $$

where $$\theta_{ijk}$$ is the bond angle between bonds $$i-j$$ and $$j-k$$.

Additionally, $$ a_{ij} $$ is given by the following expression:

$$ a_{ij} = (1+\alpha^n\eta^n_{ij})^{-1/2n} $$

where $$\eta_{ij}$$ is:

$$ \eta_{ij} = \sum_{k \neq i,j} f_c(r_{ik})\exp[\lambda_3^3(r_{ij}-r_{ik})^3] $$

$$\alpha$$ is generally small enough such that $$a_{ij} \simeq 1$$; however, for most of the work presented in the paper by Tersoff in 1988, as well as this model, $$\alpha=0$$ such that $$a_{ij}=1$$, and thus $$V_{ij}$$ reduces to the form:

$$ V_{ij} =  f_c(r_{ij})[f_R(r_{ij})+b_{ij}f_A(r_{ij})]. $$

### **Note:** 
This model driver is ported from LAMMPS, which uses a few function modifications to allow for commonly used variants of the Tersoff potential (as documented [here](http://lammps.sandia.gov/doc/pair_tersoff.html)). The differences are as follows:

$$\zeta_{ij}$$ is modified to use the factor m:

$$ \zeta_{ij} = \sum_{k \neq i,j}f_c(r_{ik})g(\theta_{ijk})\exp[\lambda_3^m(r_{ij}-r_{ik})^m] $$

Additionally, $$g(\theta)$$ used in this model driver is modified by the factor $$\gamma_{ijk}$$:

$$ g(\theta) = \gamma_{ijk}({1+ {c^2 \over d^2}- {c^2 \over [d^2+(cos\theta -cos\theta_0)^2]}}) $$

It can be noted that setting $$m = 3$$ and $$\gamma_{ijk} = 1$$ gives the original Tersoff form (from the 1988 papers) shown above.

Wiki Contributors

2021-01-28T10:32:25.004008	karls