Potentials

Read a potential from a python dict

Read a potential from json or yaml file.

from dftfit.potential import Potential
potential = Potential.from_file(filename)

DFTFIT can define very complex potentials. Unlike similar potential fitting software, DFTFIT allows any combination of potentials defined bellow. This allows a user for example to mix a ZBL and buckingham with a coulombic interaction seen here. Even though it may not make sense a user can mix a Tersoff, ZBL, Stillinger Weber, and python custom pair potential. The performance impact of mixing several potentials is almost negligible for small systems of less than 1000 atoms.

DFTFIT uses a json schema to represent any potential. To make DFTFIT optimize any float value in the potential replace the float value for with somthing similar to {"initial": 1.0, "bounds ": [2.0, 3.0]}. This tells DFTFIT that the initial guess should be 1.0 and to restrict the optimization values between 2.0 and 3.0. An example is shown here for MgO.

Note that the yaml schema is not the only way to provide a potential. json can be used to represent any DFTFIT yaml specification. Additionally they can be represented with normal python datastructures dict and list.

Bellow is a list of all the supported potentials. Soon EAM and arbitrary splines for potentials will be supported see issue 17

Two Body Potentials

Python Functions

DFTFIT allows for arbitrary python functions to be used for pair potentials. The only requirement is that you define a function named potential with the last arguments being the r that will be supplied by dftfit. All the other parameters to the function will be optimized. See this stack-overflow question if you need clarification on what numpy function vectorization is. Under the covers DFTFIT evaluates the function at a set number of points (np.linspace(cutoff[0], cutoff[1], samples)) to calculate the energies and forces (via the finite centered difference).

An example of the buckingham potential is below. You are free to import and call any functions within the block.

import numpy as np

def potential(A, p, C, r):
    return A * np.exp(-r/p) - C / (r**6)

yaml schema

pair:
 - type: python-function
   cutoff: [1.0, 10.0]
   samples: 1000
   function: |
     import numpy as np

     def potential(A, p, C, r):
         return A * np.exp(-r/p) - C / (r**6)
   parameters:
     - elements: ['Mg', 'Mg']
       coefficients: [1309362.2766468062, 0.104, 0.0]
     - elements: ['Mg', 'O']
       coefficients: [9892.357, 0.20199, 0.0]
     - elements: ['O', 'O']
       coefficients: [2145.7345, 0.3, 30.2222]

Coloumbic Interaction Potential

• lammps documentation
• example potential
• n parameters: where n is number of elements - 1

The coloumbic interaction potential has more knobs that other pair potentials to allow for how the long range interactions are integrated. Additionally a constraint allows for ensure that the total charge of the system is balanced. Complex forumla can be used e.g. LiTaO3.

\[E = \frac{C q_i q_j}{\epsilon r}\]

yaml schema

spec:
  constraint:
    charge_balance: MgO
  charge:
    Mg: 1.4
    O: -1.4
  kspace:
    type: pppm
    tollerance: 1e-5

ZBL Potential

• lammps documentation
• example potential
• 2 parameters: Z_1, Z_2

\[E_{ij} = \frac{Z_i Z_j e^2}{4 \pi \epsilon_0 r_{ij}} \phi(r_{ij}/a)\]

\[a = \frac{0.46850}{Z_i^{0.23} + Z_j^{0.23}}\]

\[\phi{x} = 0.18175e^{-3.19980x} + 0.50986e^{-0.94229x} + 0.28022e^{-0.40290x} + 0.02817e^{-0.20162x}\]

yaml schema

pair:
 - type: zbl
   cutoff: [3.0, 4.0]
   parameters:
     - elements: ['Mg', 'Mg']
       coefficients: [12, 12]
     - elements: ['Mg', 'O']
       coefficients: [12, 8]
     - elements: ['O', 'O']
       coefficients: [8, 8]

Lennard Jones Potential

• lammps documentation
• example potential
• parameters 2: ε, σ

\[E = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right]\]

yaml schema

pair:
 - type: lennard-jones
   cutoff: [10.0]
   parameters:
     - elements: ['Ne', 'Ne']
       coefficients: [33.921, 2.801]

Beck Potential

• lammps documentation
• example potential
• 5 parameters: A, B, a, α, β

\[E(r) = A \exp \left[ -\alpha r - \beta r^6 \right] -\frac{B}{(r^2 + a^2)^3} \left( 1 + \frac{2.709 + 3a^2}{r^2 + a^2} \right)\]

yaml schema

pair:
 - type: beck
   cutoff: [8.0]
   parameters:
     - elements: ['He', 'He']
       coefficients: [399.671876712, 0.0000867636112694, 0.675, 4.390, 0.0003746]

Buckingham Potential

• lammps documentation
• example potential
• 3 parameters: A, ρ, C

\[\psi(r) = A \exp^{-\frac{r}{\rho}} - \frac{C}{r^6}\]

yaml schema

pair:
 - type: buckingham
   cutoff: [10.0]
   parameters:
     - elements: ['Mg', 'Mg']
       coefficients: [1309362.2766468062, 0.104, 0.0]
     - elements: ['Mg', 'O']
       coefficients: [9892.357, 0.20199, 0.0]
     - elements: ['O', 'O']
       coefficients: [2145.7345, 0.3, 30.2222]

Three Body Potentials

Three Body potentials tend to have many more parameters. Because of this there are often mixing rules that help to reduce the number of parameters. They define some rules such that given interaction element_{i, i} \(\nabla_i\) and element_{j,j} \(\nabla_i\) the potential for intercation element_{ij} can be calculated via \(f(\nabda_i, \nabla_j)\).

Currently defined mixes:

• tersoff-2

Tersoff Potential

• lammmps documentation
• example potential
• 14 parameters: \(m, \gamma, \lambda_3, c, d, \cos(\theta_0), n, beta, \lambda_2, B, R, D, \lambda_1, A\)

yaml schema

pair:
 - type: tersoff
   parameters:
     - elements: ['C', 'C', 'C']
       coefficients: [3.0, 1.0, 0.0, 38049, 4.3484, -0.57058, 0.72751, 0.00000015724, 2.2119, 346.7, 1.95, 0.15, 3.4879, 1393.6]
     - elements: ['Si', 'Si', 'Si']
       coefficients: [3.0, 1.0, 0.0, 100390, 16.217, -0.59825 ,0.78734, 0.0000011, 1.73222, 471.18, 2.85, 0.15, 2.4799, 1830.8]
     - elements: ['Si', 'Si', 'C']
       coefficients: [3.0, 1.0, 0.0, 100390, 16.217, -0.59825, 0.0, 0.0, 0.0, 0.0, 2.36, 0.15, 0.0, 0.0]
     - elements: ['Si', 'C', 'C']
       coefficients: [3.0, 1.0, 0.0, 100390, 16.217, -0.59825, 0.787340, 0.0000011, 1.97205, 395.126, 2.36, 0.15, 2.9839, 1597.3111]
     - elements: ['C', 'Si', 'Si']
       coefficients: [3.0, 1.0, 0.0, 38049, 4.3484, -0.57058, 0.72751, 0.00000015724, 1.97205, 395.126, 2.36, 0.15, 2.9839, 1597.3111]
     - elements: ['C', 'Si', 'C']
       coefficients: [3.0, 1.0, 0.0, 38049, 4.3484, -0.57058, 0.0, 0.0, 0.0, 0.0, 1.95, 0.15, 0.0, 0.0]
     - elements: ['C', 'C', 'Si']
       coefficients: [3.0, 1.0, 0.0, 38049, 4.3484, -0.57058, 0.0, 0.0, 0.0, 0.0, 2.36, 0.15, 0.0, 0.0]
     - elements: ['Si', 'C', 'Si']
       coefficients: [3.0, 1.0, 0.0, 100390, 16.217, -0.59825, 0.0, 0.0, 0.0, 0.0, 2.85, 0.15, 0.0, 0.0]

Equations

\[E = \frac{1}{2} \sum_i \sum_{j \ne i} V_{ij}\]

\[V_{ij} = f_c\left(r_{ij}\right) \left[f_R(r_{ij}) + b_{ij} f_A(r_{ij})\right]\]

\[\begin{split}f_c(r_{ij}) = \left\{ \begin{array}{lr} 1 & r_{ij} < R_{ij} - D_{ij} \\ \frac{1}{2} - \frac{1}{2} \sin \left[\frac{\pi}{2}(r_{ij} - R_{ij})/D_{ij}\right] & R_{ij} - S_{ij} < r_{ij} < R_{ij} + D_{ij} \\ 0 & r_{ij} > R_{ij} + D_{ij} \end{array} \right.\end{split}\]

\[f_R(r) = A_{ij} \exp ( -{\lambda_{1, ij}} r )\]

\[f_A(r) = -B_{ij} \exp ( -\lambda_{2, ij} r )\]

\[b_{ij} = (1 + \beta_i^{n_i} \zeta_{ij}^{n_i})^{-\frac{1}{2{n_i}}}\]

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

\[g(\theta_{ijk}) = \gamma_{ik}\left( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{[d_i^2 + (\cos \theta_{0, i} - \cos \theta_{ijk})^2]}\right)\]

Variables: \(R_{ij}, D_{ij}, A_{ij}, \lambda_{1, ij}, B_{ij}, \lambda_{2, ij}, \beta_i, n_i, \gamma_{ik}, c_i, d_i, m_i, \lambda_{3, ij}, \theta_{0, i}\)

Two body terms (6): \(n_i, \beta_i, \lambda_{2, ij}, B_{ij}, \lambda_{1, ij}, A_{ij}\)

Three body terms (6): \(m_i, \gamma_{ik}, \lambda_{3, ij}, c_i, d_i, \theta_{0, i}\)

Terms that only depend on primary atom (6): \(n_i, \beta_i, m_i, c_i, d_i, \theta_{0, i}\)

Usually Fixed Terms $`m, gamma, beta`$

Mixing Terms \(\lambda, A, B, R, D\)

__m must be 3 or 1__

Original tersoff [1] form achieved when \(m = 3\) and \(\gamma = 1\)

Tersoff [2] has the the following contstraints:

\(\lambda_{3, i} = 0\) thus m has not effect. In original paper \(\gamma_{ik} = 1\).

Additional assumptions are the following: \(\lambda_3 = 0\), m = 3, and gamma = 1 thus these parameters are not included.

The order of the parameters are \(c, d, \cos(\theta_0), n, \beta, \lambda_2, B, R, D, \lambda_1, A\). Additional models may be added if necessary.

\[\lambda_{ij} = \frac{1}{2} (\lambda_{i} + \lambda_{j})\]

\[A_{ij} = \sqrt{A_i A_j}\]

\[B_{ij} = \chi_{ij} \sqrt{B_i B_j}\]

A mixing parameter is required for elements (N -1) see paper

\[R_{ij} = \sqrt{R_i R_j}\]

\[D_{ij} = \sqrt{D_i D_j}\]

Albe [3] when \(\beta = 1\) and \(m = 1\).

From [4] an R is 1.95, 2.85 for C-C-C and Si-Si-Si respectively and 0.15 for D (units Angstroms). R and D are chosen so as to include the first neighbor shell only.

1. Tersoff Original Paper J. Tersoff, Phys Rev B, 37, 6991 (1988).
2. Albe Form
3. Tersoff 2
4. Lammps Implementation

Stillinger Weber Potential

• lammps documentation
• example potential
• 11 parameters: \(\epsilon, \sigma, a, \lambda, \gamma, \cos(\theta_0), A, B, p, q, tol\)

yaml schema

pair:
  - type: stillinger-weber
    parameters:
      - elements: ["Cd", "Cd", "Cd"]
        coefficients: [1.03, 2.51, 1.80, 25.0, 1.20, -0.333333333333, 5.1726, 0.8807, 4.0, 0.0, 0.0]
      - elements: ["Te", "Te", "Te"]
        coefficients: [1.03, 2.51, 1.80, 25.0, 1.20, -0.333333333333, 8.1415, 0.6671, 4.0, 0.0, 0.0]
      - elements: ["Cd", "Cd", "Te"]
        coefficients: [1.03, 0.0 , 0.0, 25.0, 0.0, -0.333333333333, 0.0, 0.0, 0.0, 0.0, 0.0]
      - elements: ["Cd", "Te", "Te"]
        coefficients: [1.03, 2.51, 1.80, 25.0, 1.20, -0.333333333333, 7.0496, 0.6022, 4.0, 0.0, 0.0]
      - elements: ["Te", "Cd", "Cd"]
        coefficients: [1.03, 2.51, 1.80, 25.0, 1.20, -0.333333333333, 7.0496, 0.6022, 4.0, 0.0, 0.0]
      - elements: ["Te", "Cd", "Te"]
        coefficients: [1.03, 0.0, 0.0, 25.0, 0.0, -0.333333333333, 0.0, 0.0, 0.0, 0.0, 0.0]
      - elements: ["Te", "Te", "Cd"]
        coefficients: [1.03, 0.0, 0.0, 25.0, 0.0, -0.333333333333, 0.0, 0.0, 0.0, 0.0, 0.0]
      - elements: ["Cd", "Te", "Cd"]
        coefficients: [1.03, 0.0, 0.0, 25.0, 0.0, -0.333333333333, 0.0, 0.0, 0.0, 0.0, 0.0]

Equations

\[E = \sum_i \sum_{j > i} \phi_2(r_{ij}) + \sum_i \sum_{j \ne i} \sum_{k > j} \phi_3(r_{ij}, r_{ik}, \theta_{ijk})\]

\[\phi_2(r_{ij}) = A_{ij} \epsilon_{ij} \left[ B_{ij} \left( \frac{\sigma_{ij}}{r_{ij}} \right)^{p_{ij}} - \left( \frac{\sigma_{ij}}{r_{ij}} \right)^{q_{ij}} \right] \exp \left( \frac{\sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right)\]

\[\phi_3(r_{ij}, r_{ik}, \theta_{ijk}) = \lambda_{ijk} \epsilon_{ijk} \left[ cos \theta_{ijk} - cos \theta_{0ijk} \right]^2 exp \left( \frac{\gamma_{ij} \sigma_{ij}}{r_{ij} - a_{ij}\sigma_{ij}} \right) exp \left( \frac{\gamma_{ij} \sigma_{ik}}{r_{ik} - a_{ik}\sigma_{ik}} \right)\]

Parameters: \(\epsilon, \sigma, a, \lambda, \gamma, \cos(\theta_0), A, B, p, q, tol\)

Mixing terms: \(\sigma, \epsilon\)

### Mixing Rules

Analysis of the mixing rules for the Stillinger–Weber potential: a case-study of Ge–Si interactions in the liquid phase

https://doi.org/10.1016/j.jnoncrysol.2006.07.017

With such systems, however, there arises a problem of choosing suitable parameters for unlike-species interactions, i.e. devising \(\sigma_{ij}, \epsilon_{ij} from \sigma_i, sigma_j, \epsilon_i, \epsilon_j\) (for the two-body term) and \(\epsilon_{ijk}, \lambda_{ijk}\) from \(\epsilon_i, \epsilon_j, \lambda_i, and \lambda_j\), where i, j, and k label the species of atoms in bond pairs and triplets. The two-body parameters were usually approximated using the geometric mean for the energy parameter and the arithmetic mean for the length parameter (the so-called Lorentz–Berthelot mixing rules). This had no rigoristic justification in first principles, but was analogous to what was usually done for other potentials. - page 4233

Choosing mixed-species paramters \(\epsilon_{ijk}, \lambda_{ijk}\) for the three-body part is less obvious. Usually the choice of \(\epsilon_{ijk} = \sqrt{\epsilon_{ij}\epsilon_{ik}} = \epsilon^{\frac{1}{4}}_j \epsilon^{\frac{1}{2}}_j \epsilon^{\frac{1}{4}}_j\) and \(\lambda_{ijk} = \sqrt{\lambda_{ij}\lambda_{ik}} = \lambda^{\frac{1}{4}}_j \lambda^{\frac{1}{2}}_j \lambda^{\frac{1}{4}}_j\), first made by Grabow and Gilmer in [1] was iterated, even though the original authors had not justified it in any way.

In our study we decided to further test this traditional choice against other ways of constructing the parameters, eg. \(\lambda_{Si Si Ge} = \sqrt[3]{\lambda_{Si} \lambda_{Si} \lambda_{Ge}}\).

Since the resultant parameters differed by only a few percent, we expected to obtain similar results, regardless of the type of the mixing rule employed, which would then confirm the validity of the Grabow–Gilmer mixing as one of several that work. Surprisingly, this was not the case. It turned out that the simulations performed with only slightly different parameters resulted in radically different final atomic configurations.

1. M.H. Grabow, G.H. Gilmer, Surf. Sci. 194 (1987) 333

Gao Weber Potential

• [lammps documentation](https://lammps.sandia.gov/doc/pair_gw.html)
• [example potential](https://github.com/costrouc/dftfit/blob/master/test_files/potential/SiC-gao-weber.yaml)
• 14 parameters: $`m, gamma, lambda_3, c, d, h, n, beta ,lambda_2, B, R, D, lambda_1, A`$

yaml schema

pair:
  - type: gao-weber
    parameters:
      - elements: ['Si', 'Si', 'Si']
        coefficients: [1, 0.013318, 0, 14, 2.1, -1, 0.78000, 1, 1.80821400248640, 632.658058300867, 2.35, 0.15, 2.38684248328205, 1708.79738703139]
      - elements: ['Si', 'Si', 'C']
        coefficients: [1, 0.013318, 0, 14, 2.1, -1, 0.78000, 1, 1.80821400248640, 632.658058300867, 2.35, 0.15, 2.38684248328205, 1708.79738703139]
      - elements: ['Si', 'C', 'Si']
        coefficients: [1, 0.013318, 0, 14, 2.1, -1, 0.78000, 1, 1.96859970919818, 428.946015420752, 2.35, 0.15, 3.03361215187440, 1820.05673775234]
      - elements: ['C', 'Si', 'Si']
        coefficients: [1, 0.011304, 0, 19, 2.5, -1, 0.80468, 1, 1.96859970919818, 428.946015420752, 2.35, 0.15, 3.03361215187440, 1820.05673775234]
      - elements: ['C', 'C', 'Si']
        coefficients: [1, 0.011304, 0, 19, 2.5, -1, 0.80469, 1, 1.76776695296637, 203.208547714849, 2.35, 0.15, 2.54558441227157, 458.510465798439]
      - elements: ['C', 'Si', 'C']
        coefficients: [1, 0.011304, 0, 19, 2.5, -1, 0.80469, 1, 1.96859970919818, 428.946015420752, 2.35, 0.15, 3.03361215187440, 1820.05673775234]
      - elements: ['Si', 'C', 'C']
        coefficients: [1, 0.013318, 0, 14, 2.1, -1, 0.78000, 1, 1.96859970919818, 428.946015420752, 2.35, 0.15, 3.03361215187440, 1820.05673775234]
      - elements: ['C', 'C', 'C']
        coefficients: [1, 0.011304, 0, 19, 2.5, -1, 0.80469, 1, 1.76776695296637, 203.208547714849, 2.35, 0.15, 2.54558441227157, 458.510465798439]

Equations

Not documented see publication: Gao and Weber, Nuclear Instruments and Methods in Physics Research B 191 (2012) 504.

Vashishta Potential

• lammps documentation
• example potential
• 14 parameters: \(H, \eta, Z_i, Z_j, \lambda_1, D, \lambda_4, W, rc, B, \gamma, r_0, C, \cos(\theta)\)

yaml schema

pair:
 - type: vashishta
   parameters:
     - elements: ['C', 'C', 'C']
       coefficients: [471.74538, 7, -1.201, -1.201, 5.0, 0.0, 3.0, 0.0, 7.35, 0.0, 0.0, 0.0, 0.0, 0.0]
     - elements: ['Si', 'Si', 'Si']
       coefficients: [23.67291, 7, 1.201, 1.201, 5.0, 15.575, 3.0, 0.0, 7.35, 0.0, 0.0, 0.0, 0.0, 0.0]
     - elements: ['C', 'Si', 'Si']
       coefficients: [447.09026, 9, -1.201, 1.201, 5.0, 7.7874, 3.0, 61.4694, 7.35, 9.003, 1.0, 2.90, 5.0, -0.333333333333]
     - elements: ['Si', 'C', 'C']
       coefficients: [447.09026, 9, 1.201, -1.201, 5.0, 7.7874, 3.0, 61.4694, 7.35, 9.003, 1.0, 2.90, 5.0, -0.333333333333]
     - elements: ['C', 'C', 'Si']
       coefficients: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
     - elements: ['C', 'Si', 'C']
       coefficients: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
     - elements: ['Si', 'C', 'Si']
       coefficients: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
     - elements: ['Si', 'Si', 'C']
       coefficients: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

Equations

\[U = \sum^N_i \sum^N_{j > i} U_{ij}^{(2)}(r_{ij}) + \sum^N_i \sum^N_{j \ne i} \sum^N_{k>j,k \ne i} U_{ijk}^{(3)} (r_{ij}, r_{ik}, \theta_{ijk})\]

\[U_{ij}^{(2)}(r) = \frac{H_{ij}}{r^{eta_{ij}}} + \frac{Z_i Z_j}{r} \exp(-r/\lambda_{1, ij}) - \frac{D_{ij}}{r^4} \exp(-r/\lambda_{4, ij}) - \frac{W_{ij}}{r^6}, r<r_{c, ij}\]

\[U_{ijk}^{(3)} (r_{ij}, r_{ik}, \theta_{ijk}) = B_{ijk} \frac{[\cos \theta_{ijk} - \cos \theta_{0ijk}]^2}{1 + C_{ijk}[\cos \theta_{ijk} - \cos \theta_{0ijk}]^2} \times \exp \left( \frac{\gamma_{ij}}{r_{ij} - r_{0, ij}} \right) \exp \left( \frac{\gamma_ik}{r_{ik} - r_{0, ik}} \right), r_{ij} < r_{0, ij}, r_{ik} < r_{0, ik}\]

COMB Potential

• lammps documentation
• example comb potential
• example comb3 potential
• 46 parameters (comb), 71 parameters (comb3)

Spec not provided here because is so large. See examples.

Equations

\[E = \sum_i [E_i^{self} (q_i) + \sum_{j>i}[E_{ij}^{short}(r_{ij}, q_i, q_j) + E_{ij}^{Coul}(r_{ij}, q_i, q_j)] + E^{polar}(q_i, r_{ij}) + E^{vdw}(r_{ij}) + E^{barr}(q_i) + E^{corr}(r_{ij}, \theta_{jik})]\]

See publication for full parameter list.

• COMB - T.-R. Shan, B. D. Dvine, T. W. Kemper, S. B. Sinnott, and S. R. Phillpot, Phys. Rev. B 81, 125328 (2010)
• COMB3 - T. Liang, T.-R. Shan, Y.-T. Cheng, B. D. Devine, M. Noordhoek, Y. Li, Z. Lu, S. R. Phillpot, and S. B. Sinnott, Mat. Sci. & Eng: R 74, 255-279 (2013).