# 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