List of Potential Forms

Key to features:

The Features field in the following potential description may contain the following values:

  • potential-form - can be used in the [Pair], [EAM-Embed] and [EAM-Density] sections of a potable input file.
  • potential-function - can also be used as a function in sections of input files that accept mathematical expressions.
  • deriv - potential form provides an analytical derivative with respect to separation.
  • deriv2 - provides an analytical second derivative with respect to separation.

Born-Mayer (bornmayer)

\[V(r_{ij}) = A \exp \left( \frac{-r_{ij}}{\rho} \right)\]
potable signature:
 as.bornmayer \(A\) \(\rho\)
Features:potential-form, potential-function, deriv, deriv2

Buckingham (buck)

Potential form due to R.A. Buckingham [Buckingham1938]

\[V(r_{ij}) = A \exp \left( - \frac{r_{ij}}{\rho} \right) - \frac{C}{r_{ij}^6}\]
potable signature:
 as.buck \(A\) \(\rho\) \(C\)
Features:potential-form, potential-function, deriv, deriv2.

Buckingham-4 (buck4)

Four-range Buckingham potential due to B. Vessal et al. [Vessal1989], [Vessal1993].

\[\begin{split}V(r_{ij}) = \begin{cases} A \exp(-r_{ij}/\rho) , & 0 \leq r_{ij} \leq r_\text{detach}\\ a_0 + a_1 r_{ij} +a_2 r_{ij}^2+a_3 r_{ij}^3+a_4 r_{ij}^4+a_5 r_{ij}^5, & r_\text{detach} < r_{ij} < r_\text{min}\\ b_0 +b_1 r_{ij}+b_2 r_{ij}^2+b_3 r_{ij}^3 , & r_\text{min} \leq r_{ij} < r_\text{attach}\\ -\frac{C}{r_{ij}^6} , & r_{ij} \geq r_\text{attach}\\ \end{cases}\end{split}\]

In other words this is a Buckingham potential in which the Born-Mayer component acts at small separations and the disprsion term acts at larger separation. These two parts are linked by a fifth then third order polynomial (with a minimum formed in the spline at \(r_\text{min}\)).

The spline parameters (\(a_{0\ldots 5}\) and \(b_{0\ldots 3}\)) are subject to the constraints that \(V(r_{ij})\), first and second derivatives must be equal at the boundary points and the function must have a stationary point at \(r_{min}\). The spline coefficients are automatically calculated by this potential-form.

Note

Due to the complexity of calculating the spline-coefficients this potential form does not have an equivalent in the atsim.potentials.potentialfunctions module.

potable signature:
 as.buck4 \(A\) \(\rho\) \(C\) \(r_\text{detach}\) \(r_\text{min}\) \(r_\text{attach}\)
Features:potential-form, deriv, deriv2

Constant (constant)

Potential form that always evaluates to a constant value.

\[V(r_{ij}) = C\]
potable signature:
 as.constant C
Features:potential-form, potential-function, deriv, deriv2

Coulomb (coul)

Electrostatic interaction between two point charges.

\[V(r_{ij}) = \frac{ q_i q_j }{4 \pi \epsilon_0 r_{ij} }\]

Note

Constant value appropriate for \(r_{ij}\) in angstroms and energy in eV.

potable signature:
 as.coul \(q_i\) \(q_j\)
Features:potential-form, potential-function, deriv, deriv2

Exponential (exponential)

General exponential form.

\[V(r_{ij}) = A r_{ij}^n\]
potable signature:
 as.exponential \(A\) \(n\)
Features:potential-form, potential-function, deriv, deriv2

Exponential Spline (exp_spline)

Exponential spline function (as used in splining routines).

\[V(r_{ij}) = \exp \left( B_0 + B_1 r_{ij} + B_2 r_{ij}^2 + B_3 r_{ij}^3 + B_4 r_{ij}^4 + B_5 r_{ij}^5 \right) + C\]

Where \(B_0\), \(B_1\), \(B_2\), \(B_3\), \(B_4\), \(B_5\), \(C\) are spline coefficients.

potable signature:
 as.exp_spline \(B_0\) \(B_1\) \(B_2\) \(B_3\) \(B_4\) \(B_5\) \(C\)
Features:potential-form, potential-function, deriv, deriv2

Hydrogen Bond 12-10 (hbnd)

\[V(r_{ij}) = \frac{A}{r_{ij}^{12}} - \frac{B}{r_{ij}^{10}}\]
portable signature:
 as.hbnd \(A\) \(B\)
Features:potential-form, potential-function, deriv, deriv2

Lennard-Jones 12-6 (lj)

Potential form first proposed by John Lennard-Jones in 1924 [Lennard-Jones1924].

\[V(r_{ij}) = 4 \epsilon \left( \frac{\sigma^{12}}{r_{ij}^{12}} - \frac{\sigma^6}{r_{ij}^6} \right)\]

\(\epsilon\) defines depth of potential well and \(\sigma\) is the separation at which \(V(r_{ij})\) is zero.

potable signature:
 as.lj \(\epsilon\) \(\sigma\)
Features:potential-form, potential-function, deriv, deriv2

Morse (morse)

\[V(r_{ij}) = D \left[ \exp \left( -2 \gamma (r_{ij} - r_*) \right) - 2 \exp \left( -\gamma (r - r_*) \right) \right]\]

\(-D\) is the potential well depth at an equilibrium separation of \(r_*\).

potable signature:
 as.morse \(\gamma\) \(r_*\) \(D\)
Features:potential-form, potential-function, deriv, deriv2

Polynomial (polynomial)

Polynomial of arbitrary order.

\[V(r_{ij}) = C_0 + C_1 r_{ij} + C_2 r_{ij}^2 + \dots + C_n r_{ij}^n\]

This function accepts a variable number of arguments which are \(C_0, C_1, \dots, C_n\) respectively.

potable signatures:
 as.polynomial \(C_0 ... C_n\)
Features:potential-form, potential-function, deriv, deriv2

Square Root (sqrt)

Potential function is:

\[U(r_{ij}) = G\sqrt{r_{ij}}\]
potable signature:
 as.sqrt \(G\)
Features:potential-form, potential-function, deriv, deriv2

Tang-Toennies (tang_toennies)

This potential form was derived to describe the Van der Waal’s interactions between the noble gases (He to Rn) by Tang and Toennies [Tang2003].

This has the following form:

\[V(r) = A \exp(-br) - \sum_{n=3}^N f_{2N} (bR) \frac{C_{2N}}{R^{2N}}\]

Where:

\[f_{2N}(x) = 1- \exp(-x) \sum_{k=0}^{2n} \frac{x^k}{k!}\]
potable signature:
 as.tang_toennies \(A\) \(b\) \(C_6\) \(C_8\) \(C_{10}\)
Features:potential-form, potential-function, deriv, deriv2

Zero (zero)

Potential form which returns zero for all separations.

\[V(r) = 0\]
potable signature:
 as.zero
Features:potential-form, potential-function, deriv, deriv2

Ziegler-Biersack-Littmark (zbl)

Ziegler-Biersack-Littmark screened nuclear repulsion for describing high energy interactions [Ziegler2015].

\[\begin{split}V(r) & = \; & \frac{1}{4\pi\epsilon_0} \frac{Z_1}{Z_2} \phi(r/a) + S(r) \\ a & = & \frac{0.46850}{Z_i^{0.23} + Z_j^{0.23}} \\ \phi(x) & = & 0.18175 \exp(-3.19980x) \\ & & \quad + 0.50986 \exp(-0.94229x) \\ & & \quad + 0.28022\exp(-0.40290x) \\ & & \quad + 0.02817\exp(-0.20162x) \\\end{split}\]

Where \(Z_i\) and \(Z_j\) are the atomic numbers of two species.

potable signature:
 as.zbl \(Z_i\) \(Z_j\)
Features:potential-form, potential-function, deriv, deriv2