Three-body cluster potential by Stephenson, Radny and Smith (1996) v000

Ben Druecke; Daniel S. Karls; Philip C. L. Stephenson; Marian W. Radny; Philip V. Smith

doi:10.25950/f9b3bf83

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ThreeBodyCluster_SRS__MD_814842199451_000

Title A single sentence description.	Three-body cluster potential by Stephenson, Radny and Smith (1996) v000
Description	A modification of the Stillinger-Weber three-body cluster potential designed for improved surface properties.
Disclaimer A statement of applicability provided by the contributor, informing users of the intended use of this KIM Item.	None

Contributor	Ben Druecke
Maintainer	Ben Druecke
Implementer	Ben Druecke Daniel S. Karls
Developer	Philip C. L. Stephenson Marian W. Radny Philip V. Smith
Published on KIM	2019
How to Cite	This Model Driver originally published in [1] is archived in OpenKIM [2-4]. [1] Stephenson PCL, Radny MW, Smith PV. A modified Stillinger-Weber potential for modelling silicon surfaces. Surface Science. 1996;366(1):177–84. doi:10.1016/0039-6028(96)00801-1 [2] Druecke B, Karls DS, Stephenson PCL, Radny MW, Smith PV. Three-body cluster potential by Stephenson, Radny and Smith (1996) v000. OpenKIM; 2019. doi:10.25950/f9b3bf83 [3] Tadmor EB, Elliott RS, Sethna JP, Miller RE, Becker CA. The potential of atomistic simulations and the Knowledgebase of Interatomic Models. JOM. 2011;63(7):17. doi:10.1007/s11837-011-0102-6 [4] Elliott RS, Tadmor EB. Knowledgebase of Interatomic Models (KIM) Application Programming Interface (API). OpenKIM; 2011. doi:10.25950/ff8f563a
Funding	Not available

Short KIM ID The unique KIM identifier code.	MD_814842199451_000
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.	ThreeBodyCluster_SRS__MD_814842199451_000
DOI	10.25950/f9b3bf83 https://doi.org/10.25950/f9b3bf83 https://commons.datacite.org/doi.org/10.25950/f9b3bf83
KIM Item Type	Model Driver
KIM API Version	2.0
Programming Language(s) The programming languages used in the code and the percentage of the code written in each one.	100.00% C

Models using this Model Driver

ThreeBodyCluster_SRS_StephensonRadnySmith_1996_Si__MO_604248666067_000

The [Stephenson-Radny-Smith potential](https://www.sciencedirect.com/science/article/pii/0039602896008011) (SRS1996) is a three-body interatomic potential originally developed to model free surface behavior of silicon. It is a generalized form of the classic [Stillinger-Weber potential](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.5262) with four additional parameters to allow for better tuning of free surface behavior. It was originally created to better match experimental observations and ab initio calculations of bond lengths and bond energies on the $$7 \times 7$$ unit cell of the (111)Si free surface.

## Parameters
The model has eleven dimensionless parameters ($$A$$, $$B$$, $$p$$, $$q$$, $$a$$, $$\lambda$$, $$\gamma$$, $$\zeta$$, $$b$$, $$k$$ and $$c$$) and two characteristic scales ($$\sigma$$ the characteristic interatomic spacing, and $$\epsilon$$ the characteristic energy).

## Function
The generic model of potential energy for a three-body potential composed of $$N$$ atoms is

$$\mathcal{V}_{(N)}^{\mathrm{int}} = \epsilon \phi_{0} + \frac{1}{2!}\sum_{\begin{array}{c}i,j\\i\neq j\end{array}}^{N}{\epsilon \phi_{2} \left( \frac{r_{ij}^{*}}{\sigma} \right)}
+ \frac{1}{3!}\sum_{\begin{array}{c}i,j,k\\i\neq j \neq k\end{array}}^{N}{\epsilon \phi_{3} \left( \frac{r_{ij}^{*}}{\sigma}, \frac{r_{jk}^{*}}{\sigma}, \frac{r_{ki}^{*}}{\sigma} \right)}$$

where $$\phi_{0}$$ accounts for the number of particles in the system but is independent of their positions, $$\phi_{2}$$ is the pairwise potential and $$\phi_{3}$$ the three-body potential. The dimensional lengths, $$r_{ij}^{*}$$, are non-dimensionalized with the charactersitic interatomic spacing $$\sigma$$, and the energies are non-dimensionalized with the characteristic bond energy $$\epsilon$$.

For SRS1996 (as well as the classic Stillinger-Weber potential), the pairwise term is

$$
\phi_{2}(r_{ij}) = \left\{
\begin{array}{lr}
\displaystyle
A \left( \frac{B}{r_{ij}^{p}} - \frac{1}{r_{ij}^{q}} \right) \exp \left( \frac{\zeta}{r_{ij} - a} \right), & r_{ij} \le a
\\
0,& \mathrm{otherwise}
\end{array}
\right.
$$

where $$r_{ij} = r_{ij}^{*}/\sigma$$ is the dimensionless distance between atoms $$i$$ and $$j$$.  For the pairwise term, the only difference between SRS1996 and Stillinger-Weber is in the numerator of the exponential function, which SRS1996 parameterizes as $$\zeta$$ and Stillinger-Weber uses a value of one.

The functional form of the three-body potential for SRS1996 is

$$
\phi_{3}(r_{ij},r_{jk},r_{ki}) = h\left(r_{ij},r_{ik},\theta_{jik} \right)
+ h\left(r_{ij},r_{kj},\theta_{ijk} \right)
+ h\left(r_{ik},r_{kj},\theta_{ikj} \right)
$$

where

$$
h(r,s,\theta) = \left\{ \begin{array}{lr}
\displaystyle
\lambda \left( b \left( \cos \theta + k \right)^{2} - c \right) \exp \left( \gamma \left( \frac{1}{r - a} + \frac{1}{s-a} \right) \right),
&r ~ \mathrm{and} ~ s \le a
\\
0,&  \mathrm{otherwise}
\end{array} \right.
$$

and $$\theta_{ijk}$$ is the angle between the position vectors $$\mathbf{r}_{ji}$$ and $$\mathbf{r}_{jk}$$. This differs from Stillinger-Weber in the use of three additional parameters ($$b$$, $$k$$ and $$c$$) for the angular dependence. (The Stillinger-Weber formulation is obtained by setting $$b = 1$$, $$k = 1/3$$ and $$c = 0$$.) The use of these additional parameters in this model allows more flexibility in tuning the bond angles for modeling the behavior of free surfaces. Specifically the value $$k$$ determines the bond angle with the minimum energy according to

$$
\theta_{\mathrm{min}} = \cos^{-1} \left( -k \right).
$$

Wiki Contributors

2019-06-03T08:43:33.038546	bdruecke
2019-05-28T17:29:04.926294	bdruecke
2019-05-28T17:27:29.620610	bdruecke
2019-05-28T17:25:40.970316	bdruecke

ThreeBodyCluster_SRS__MD_814842199451_000

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