Environment-Dependent Interatomic Potential (EDIP) model driver v002

Daniel S. Karls; Joao F. Justo; Martin Z. Bazant; Efthimios Kaxiras; Vasily V Bulatov; Sidney Yip

doi:10.25950/75c4686e

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EDIP__MD_506186535567_002

Title A single sentence description.	Environment-Dependent Interatomic Potential (EDIP) model driver v002
Description	This model driver is a C-based implementation of the three-body Environment Dependent Interatomic Potential (EDIP) developed by Martin Bazant and Efthemios Kaxiras and published in 1996, a bond-order potential primarily used to simulate a wide variety of structures and defects associated with silicon, carbon, and germanium. The implementation itself is based on version 1.1 of the C code published by Bazant on http://web.mit.edu/bazant/www/EDIP/index.html. All functional forms are evaluated analytically, i.e. no splines are used for evaluation.
Disclaimer A statement of applicability provided by the contributor, informing users of the intended use of this KIM Item.	None

Contributor	Daniel S. Karls
Maintainer	Daniel S. Karls
Implementer	Daniel S. Karls
Developer	Joao F. Justo Martin Z. Bazant Efthimios Kaxiras Vasily V Bulatov Sidney Yip
Published on KIM	2018
How to Cite	This Model Driver originally published in [1-3] is archived in OpenKIM [4-6]. [1] Bazant MZ, Kaxiras E. Modeling of Covalent Bonding in Solids by Inversion of Cohesive Energy Curves. Physical Review Letters. 1996Nov;77(21):4370–3. doi:10.1103/PhysRevLett.77.4370 [2] Bazant MZ, Kaxiras E, Justo JF. Environment-dependent interatomic potential for bulk silicon. Physical Review B. 1997Oct;56(14):8542–52. doi:10.1103/PhysRevB.56.8542 [3] Justo JF, Bazant MZ, Kaxiras E, Bulatov VV, Yip S. Interatomic potential for silicon defects and disordered phases. Physical Review B. 1998Aug;58(5):2539–50. doi:10.1103/PhysRevB.58.2539 [4] Karls DS, Justo JF, Bazant MZ, Kaxiras E, Bulatov VV, Yip S. Environment-Dependent Interatomic Potential (EDIP) model driver v002. OpenKIM; 2018. doi:10.25950/75c4686e [5] 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 [6] Elliott RS, Tadmor EB. Knowledgebase of Interatomic Models (KIM) Application Programming Interface (API). OpenKIM; 2011. doi:10.25950/ff8f563a Click here to download the above citation in BibTeX format.
Funding	Not available

Short KIM ID The unique KIM identifier code.	MD_506186535567_002
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.	EDIP__MD_506186535567_002
DOI	10.25950/75c4686e https://doi.org/10.25950/75c4686e https://commons.datacite.org/doi.org/10.25950/75c4686e
KIM Item Type	Model Driver
KIM API Version	2.0

Previous Version	EDIP__MD_506186535567_001

Models using this Model Driver

EDIP_BelkoGusakovDorozhkin_2010_Ge__MO_129433059219_001
EDIP_JustoBazantKaxiras_1998_Si__MO_958932894036_002

Files

CMakeLists.txt
EDIP.c
LICENSE.CDDL
README
kimprovenance.edn
kimspec.edn

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EDIP__MD_506186535567_002.txz	Tar+XZ	Linux and OS X archive
EDIP__MD_506186535567_002.zip	Zip	Windows archive

Wiki

## Description
This **Model Driver** implements the original EDIP potential of Bazant and Kaxiras (used for silicon in its seminal publication).

## Parameters
Symbols:

$$a, A, B, \rho, \sigma, \lambda, \gamma, b, c, \mu, Q_0, \eta, \beta, \alpha, u_1, u_2, u_3, u_4$$

Corresponding variables in code:
a, A, B, rh, sig, lam, gam, b, c, mu, Qo, eta, bet, alp, u1, u2, u3, u4

## Details

The total potential energy of a system of $$N$$ atoms is assumed to take the form $$E = \sum_{i=1}^N E_i $$, where the contribution to the energy of atom $$ i $$ is given by

$$E_i = \sum_{j \neq i} V_2(R_{ij},Z_i) + \sum_{j \neq i} \sum_{k \neq i, k > j} V_3 (\vec{R}_{ij},\vec{R}_{ik},Z_i) .$$

The "environmental dependence" of the energy (and, consequently, its derivatives) is introduced through the coordination $$Z$$, which takes the form

$$Z_i = \sum_{m \neq i} f(R_{im}) $$

where

$$ f(r) = \begin{cases}
      1 &  \text{if  } r < c\\
      \exp{\left(\dfrac{\alpha}{1-x^{-3}}\right)} & \text{if  } c < r < a\\
      0 & \text{if  } r > a
    \end{cases} $$

and $$x = \dfrac{( r-c )}{(a-c)} $$.
The two-body term $$V_2$$ of the energy includes an attractive part and a repulsive part:

$$V_2(r,Z) = A \left[ \left(\dfrac{B}{r}\right)^{\rho} - p(Z) \right]\exp\left(\dfrac{\sigma}{r-a}\right)$$

with the bond-order function taking the form $$p(Z) = e^{-\beta Z^2}$$.

The three-body contribution $$V_3$$ to the energy consists of a product of radial and angular functions:

$$V_3(\vec{R}_{ij}, \vec{R}_{ik}, Z_{i}) = g(R_{ij})g(R_{ik}) h(l_{ijk},Z_i) $$

where the bond angle of triplet $$ijk$$ is contained in $$l_{ijk} = \cos \theta_{ijk} = \vec{R}_{ij} \cdot \vec{R}_{ik} / R_{ij} R_{ik}$$.  The radial contribution is given by the function $$g(r) = \exp\left(\dfrac{\gamma}{r-a}\right)$$.  The angular contribution to $$V_3$$ is given by

$$h(l,Z) = \lambda \left[ \left( 1 - e^{-Q(Z)(l+\tau(Z))^2}\right)  + \eta Q(Z)\left(l+\tau(Z)\right)^2\right]$$

with the sensitivity of the three-body energy to coordination controlled by the function $$Q(Z) = Q_0 e^{-\mu Z}$$.  The function $$\tau(Z)$$, which controls the equilibrium angle of the three-body interaction, is chosen heuristically to have the form

$$\tau(Z) = u_1 + u_2 \left( u_3 e^{-u_4 Z} - e^{-2 u_4 Z} \right).$$

## Performance
As mentioned in the description, this model driver evaluates all of its constituent functions analytically rather than using splines.  At the cost of some numerical precision compared to analytical evaluation, those interested in optimizing performance are urged to consider using the other EDIP model driver, [EDIP_LAMMPS__MD_783584031339_000](https://doi.org/10.25950/a6a67b9f), which uses splines to evaluate its functions.

The Tunable Intrinsic Ductility Potential (TIDP) of Rajan, Warner and Curtin is based on standard Morse potential. The ductility is tuned by altering the tail of $\varphi(r)$ while leaving the energy well unchanged. The functional form is

\[\varphi(r)= \begin{cases} (1-\exp[-\alpha(r-1)])^2-1 & r \le r_1 \\ A_1 r^3 + B_1 r^2 + C_1 r + D_1 & r_1 < r \le r_2 \\ A_2 r^3 + B_2 r^2 + C_2 r + D_2 & r_2 < r \le r_3 \\ 0 & r_3 < r \end{cases}\]

The TIDP model has 12 parameters:

\[\alpha, \quad r_1, \quad r_2, \quad r_3, \quad A_1, \quad B_1, \quad C_1, \quad D_1, \quad A_2, \quad B_2, \quad C_2, \quad D_2.\]

Wiki Contributors

2024-05-07T15:03:32.809053419	karls
2024-03-29T18:00:03.663578575	karls
2021-08-24T14:42:47.145030	karls